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The method of Lagrange multipliers finds critical points (including maxima and minima) of a differentiable function subject to differentiable constraints.
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How to choose between dual gradient descent and the method of Lagrangian multipliers?
For an optimization problem
$$
\max f(x)\\\
s.t. g(x)\le 0
$$
The Lagrangian is
$$
\mathcal L(x, \lambda)=f(x)-\lambda g(x)
$$
Dual gradient descent solves it by (according to Page 43 of this lecture …